TR-2003-06

The d-dimensional rigidity matroid of sparse graphs

Abstract

Let ${\cal R}_d(G)$ be the
$d$-dimensional rigidity matroid for
a graph $G=(V,E)$. For $X\subseteq V$ let $i(X)$ be the number of edges
in the subgraph of $G$ induced by $X$.
We derive a min-max formula which determines
the rank function in ${\cal R}_d(G)$ when $G$ has
maximum degree at most
$d+2$ and minimum degree at most $d+1$.
We also show
that if $d$ is even and $i(X)\leq \frac{1}{2}[(d+2)|X| -(2d+2)]$
for all $X\subseteq V$ with $|X|\geq 2$
then $E$ is independent in ${\cal R}_d(G)$.
We conjecture that the latter result
holds for all $d\geq 2$ and prove this for the special case when
$d=3$. We use the
independence result for even $d$ to show that if the connectivity of
$G$ is sufficiently large in comparison to $d$ then
$E$ has large rank in ${\cal R}_{d}(G)$.
We use the case $d=4$ to show that, if
$G$ is $10$-connected, then $G$ can be made
rigid in ${\mathbb{R}}^{3}$ by pinning down approximately
three quarters of its vertices.